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Miloslav Ciz 2024-02-12 20:36:38 +01:00
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mandelbrot_set.md
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@ -37,7 +37,7 @@ Mandelbrot set is a famous two dimensional [fractal](fractal.md), a [set](set.md
*z[0] = 0*, *z[n + 1] = z[n]^2 + p*
Mandelbrot set is the set of all points *p* for which the [absolute value](abs.md) ("length") of *z[n]* does NOT grow towards infinity as *n* goes towards infinity.
Mandelbrot set is the set of all points *p* for which the [absolute value](abs.md) ("length") of *z[n]* stays bounded (i.e. doesn't grow beyond any limits) as *n* goes towards infinity.
I.e. taking any point *p* in the complex plane (whose real and imaginary parts we see as the *x* and *y* coordinates), plugging it into the above equation and iterating the series infinitely many times, if the absolute value of *z[n]* stays bounded under some finite value (even very large, just not infinitely large), the number belongs to the set, otherwise not (if the absolute value diverges towards infinity). I.e. in other words the Mandelbrot set is a set of kind of "well behaved" points that don't shoot away to infinity when we keep applying some operation to them over and over. Of course computers cannot evaluate infinitely many iterations of the series so they cannot compute the set 100% accurately, but we may very well [approximate](approximation.md) by performing many iterations (let's 100000) and seeing if the value we get is "very large" (let's say 1000000000) when we stop -- this will work correctly for most points and those few points near the set borders where we make a wrong guess won't really be noticed unless we zoom in very close -- in such cases we can simply perform more iterations to increase precision. To add **[colors](color.md)** to the visualization (so that we don't observe just the borders but also some kind of structure inside and outside of the set) we may simply assign different colors to the points depending e.g. on how big the absolute value is at the time we stop the evaluation, or how many iterations it took for the absolute value to exceed given limit (for points outside the set). Also note that for nice pictures we should apply [antialiasing](antialiasing.md). Additional fancy filters and [shaders](shader.md) such as some kind of postprocessing or fake 3D can also be applied to make the result even more impressive.